Questions tagged [compactness]
The compactness tag is for questions about compactness and its many variants (e.g. sequential compactness, countable compactness) as well locally compact spaces; compactifications (e.g. one-point, Stone-Čech) and other topics closely related to compactness. This includes logical compactness.
6,565 questions
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Is a compact subspace of a quotient of a topological vector space contained in the image of a compact subspace of the original space?
Question: Let $X$ be a Hausdorff topological vector space. Let $A \subseteq X$ be a closed subspace. Then the quotient $\newcommand\quotient[2]{{^{\Large #1}}/{_{\Large #2}}} \quotient{X}{A}$ is again ...
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How fast can I guarantee convergence on a compact interval?
Suppose we have a sequence $(x_n)_n\subset [a,b]$, we know by compactness there must be at least some limit point, i.e. there exists a subsequence $n_k$ and $x\in[a,b]$ such that
$$x_{n_k}\xrightarrow{...
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Order in an open set
I need help to prove a theorem.
In the following definition $\operatorname{fr}_X(A)$ means the boundary of $A$ in $X$
Definition. Let $(X, \tau_X)$ be a topological space, let $p \in X$, and let $\...
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Is every CGWH space sober?
Definition 1: A topological space is called compactly generated if it has the following property: Let $U \subseteq X$ be a subset such that for every compact Hausdorff space $K$ and every continuous ...
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Compact operators in Hilbert space
I have tried to solve this exercise:
Let $H$ be a Hilbert space and let $T = T^* \in B_{\infty}(H)$ a compact operator. Given $\psi_0 \in H$ consider the equations:
\begin{align*}
(1)\quad &T\psi =...
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Is the quotient of the subspace homeomorphic to the subspace of the quotient?
Suppose $X$ is compact Hausdorff and $\mathcal P$ is a partition of $X$. Define the map $\pi:X \to \mathcal P$ taking each point to the unique partition element containing it. Give $\mathcal P$ the ...
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Order $\omega$ in a dendrite
before I begin, I would like to provide some definitions and theorems.
Definition. A topological space $(X, \tau_X)$ is a continuum, if $X$ is a non-empty, metric, compact and connected space.
Here $\...
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Sections with compact support commutes with tensor over a locally compact Hausdorff space. Why can we reduce to the compact case?
I have a question about the proof of the following result in Kashiwara, Schapira, Sheaves on Manifolds:
Proposition 2.5.12 [Let $X$ be a Hausdorff and locally compact space.] Let $A$ be a ring, and ...
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When compacts generate the Borel $\sigma$-algebra, is $X$ $\sigma$-compact?
Let $X$ be a Hausdorff topological space and let $\mathcal K$ denote the family of compact subsets of $X$. Assume that the Borel $\sigma$-algebra $\mathcal B(X)$ is generated by compact sets, i.e.
$$
\...
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Heine-Borel Theorem in Complete Riemannian Manifolds [closed]
Let $(M,g)$ be a complete Riemannian manifold and let $d_g:M^2\to\mathbb R$ be the length-minimizing metric induced by $g$.
Does it necessarily hold that the compact subsets of $M$ are closed and $d_g$...
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When do linear constraints form a compact? [closed]
Given a list of constraints
$$
F_i = \{ x\in\mathbb{R}^n\mid L_i(x) \leq c_i \}
$$
where $L_i\in L(\mathbb{R}^n,\mathbb{R})$ and $c_i\in\mathbb{R}$ for every $i\in\{ 0,\dots,m \}$ with $m \geq n$, ...
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Uniform Compactness Radius for Locally Compact Metric Spaces
I was reading a nice paper, but right at the beginning of one of the main results they claim:
$X$ is a locally compact metric space, so it has an equivalent metric such that every closed ball of ...
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Proving every open cover having finite subcover entails sequential compactness
I am trying to prove that if it is true for a set $K$ that every open cover of it has some finite subcovering, then $K$ is sequentially compact and am looking to verify steps that I'm unsure about in ...
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Exercise on Alexandroff compactification
Let $X,Y$ be Hausdorff and locally compact topological spaces, with $Y$ not compact, and let $\hat{Y}$ be the Alexandroff compactification of $Y$. Let $A\subset X$ be an open, relatively compact ...
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Attribution for Metrization Theorem
Let $f:X \rightarrow Y$ be a continuous surjection with $X$ a compact metric space and $Y$ a Hausdorff space. Then $Y$ is metrizable.
Does this theorem have a name? Who first proved it? Just ...
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